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Quantum Harmonic Oscillator in the Path Integral Formulation and its relation to the Second Quantization

Kingshuk Sarkar

Published Apr 30, 2026• 0 citations
OpenAlex

Abstract

This thesis investigates the quantum harmonic oscillator (QHO) through the path integral formulation. It begins with a review of classical dynamics and the standard operator approach to establish a foundational framework. The core of the work involves the systematic construction and evaluation of the QHO propagator, demonstrating how the transition amplitude naturally separates into a classical contribution and quantum fluctuations. Finally, the discussion extends to systems with infinite degrees of freedom, illustrating how the oscillator structure serves as a bridge to quantum field theory and the emergence of second quantization.

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This thesis investigates the quantum harmonic oscillator (QHO) through the path integral formulation. It begins with a review of classical dynamics and the standard operator approach to establish a foundational framework. The core of the work involves the systematic construction and evaluation of the QHO propagator, demonstrating how the transition amplitude naturally separates into a classical contribution and quantum fluctuations. Finally, the discussion extends to systems with infinite degrees of freedom, illustrating how the oscillator structure serves as a bridge to quantum field theory and the emergence of second quantization.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to non-commuting operators. Generally, the quantum mechanics based on the Schrodinger equation is preferred for a system of particles. But path integral formulation of quantum mechanics finds applications in more complicated situations, particularly, in quantum field theory. The purpose of solving the Schrodinger equation lies in finding the time evolution operator which is able to generate the time translation of the given system. Richard Feynman noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. Dissipative phenomena with a quantum origin manifest themselves in many different situations and hence the study of dissipative systems and their quantization is of a great theoretical and practical value. Many dissipative systems can be modelled as a linearly damped harmonic oscillator.
Quantum field theories are quantum mechanical systems with an infinite number of degrees of freedom. We start this book with a review of quantum mechanics in a form that is particularly suited for the field theory generalization. This form utilizes the Heisenberg picture and the functional integral representation of probability amplitudes — the transformation functions of Dirac. Some time will be spent developing and discussing the functional or path integral representation of quantum mechanics since this will enable us to learn a new formalism in a simple and familiar physical context — we won't have to learn too many new things all at once. The functional formulation will be illustrated by applying it at some length to the harmonic oscillator which is not only the simplest (and exactly soluble) non-trivial quantum system but which is also the dynamics of a single mode of a free field theory. The purpose of this review of quantum mechanics is to introduce in the simplest form much of the notation and methods used later in the book. Throughout this chapter we will explicitly work only with systems that have a single degree of freedom (one coordinate variable). The extension to systems with several degrees of freedom is straightforward.
The harmonic oscillator represents as a unique tool in modern physics since it is not only exactly solvable but also relates many phenomena. This paper introduces the relation between Quantum Field Theory and other physical theories and discusses how harmonic oscillator behave in Quantum Mechanics and Quantum Field Theory by the quantization of Klein-Gordon Field. To this end, this paper applies energy-momentum equation given by special relativity to Schrödinger’s Equation to get the Klein-Gordon Field, and then uses canonical method to quantize the field, which is called the second quantization of the field. The comparison is made between the result quantized field, the quantum harmonic oscillator and classical harmonic oscillator. The similarity of two physical systems is demonstrate by the comparison between commutation relations of operators. The similar commutation relations then help people to interpret quantized Klein-Gordon Field as a system that contain a quantized harmonic oscillator in each space coordinate.
Abstract Functional integrals are basic tools to study first quantum mechanics (QM), and quantum field theory (QFT). The path integral formulation of QM is well suited to the study of systems with an arbitrary number of degrees of freedom. It makes a smooth transition between nonrelativistic QM and QFT possible. The Euclidean functional integral also emphasizes the deep connection between QFT and the statistical physics of systems with short-range interactions near a continuous phase transition. The path integral representation of the matrix elements of the quantum statistical operator e-β H for Hamiltonians of the simple separable form p2/2m +V(q) is derived. To the path integral corresponds a functional measure and expectation values called correlation functions, which are generalized moments, and related to quantum observables, after an analytic continuation in time. The path integral corresponding to the Euclidean action of a harmonic oscillator, to which is added a time-dependent external force, is calculated explicitly. The result is used to generate Gaussian correlation functions and also to reduce the evaluation of path integrals to perturbation theory. The path integral also provides a convenient tool to derive semi-classical approximations.
This chapter introduces the path integral formulation of quantum mechanics as a rigorous and systematic alternative to canonical quantization. It begins by defining the transition probability amplitude (propagator) and deriving the path integral by discretizing time and summing over all possible intermediate paths between initial and final spacetime points. The chapter shows the final form of the propagator as a functional integral over all paths, weighted by the exponential of the action. This formulation provides an intuitive understanding that paths near the classical trajectory (which minimizes the action) interfere constructively, dominating the overall amplitude.